Related papers: Fifty Three Matrix Factorizations: A systematic ap…
In this survey paper we study parametric versions of writing a matrix in $SL_n (\mathbb{C})$ as a product of lower and upper unitriangular matrices in interchanging order as well as generalizations to other classical groups. We give an…
We present new results on Boolean matrix factorization and a new algorithm based on these results. The results emphasize the significance of factorizations that provide from-below approximations of the input matrix. While the previously…
Matrix factorization exploits the idea that, in complex high-dimensional data, the actual signal typically lies in lower-dimensional structures. These lower dimensional objects provide useful insight, with interpretability favored by sparse…
Since its introduction in 2012, the factorization theory for rational motions quickly evolved and found applications in theoretical and applied mechanism science. We provide an accessible introduction to motion factorization with many…
We introduce the $D$-decomposition, a non-orthogonal matrix factorization of the form $A \approx P D Q$, where $P \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{k \times k}$, and $Q \in \mathbb{R}^{k \times n}$. The decomposition is…
As deep learning (DL) techniques become integral to various applications, ensuring model fairness while maintaining high performance has become increasingly critical, particularly in sensitive fields such as medical diagnosis. Although a…
Recently, convex formulations of low-rank matrix factorization problems have received considerable attention in machine learning. However, such formulations often require solving for a matrix of the size of the data matrix, making it…
Currently, matrix decomposition is one of the most widely used collaborative filtering algorithms by using factor decomposition to effectively deal with large-scale rating matrix. It mainly uses the interaction records between users and…
Low-rank matrix factorization (MF) is an important technique in data science. The key idea of MF is that there exists latent structures in the data, by uncovering which we could obtain a compressed representation of the data. By factorizing…
This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization…
RSVDPACK is a library of functions for computing low rank approximations of matrices. The library includes functions for computing standard (partial) factorizations such as the Singular Value Decomposition (SVD), and also so called…
We exploit the truncated singular value decomposition and the recently proposed circulant decomposition for an efficient first-order approximation of the multiplication of large dense matrices. A decomposition of each matrix into a sum of a…
This paper explores the relationship between matrix factorizations and linear matrix equations. It shows that every matrix factorization defines two hidden projectors, one for the column space and one for the row space of a matrix, and how…
We revisit a singular value decomposition (SVD) algorithm given in Chen et al. (2019b) for exploratory Item Factor Analysis (IFA). This algorithm estimates a multidimensional IFA model by SVD and was used to obtain a starting point for…
Singular value decomposition (SVD) is one of the most popular compression methods that approximate a target matrix with smaller matrices. However, standard SVD treats the parameters within the matrix with equal importance, which is a simple…
Distributions measured in high energy physics experiments are usually distorted and/or transformed by various detector effects. A regularization method for unfolding these distributions is re-formulated in terms of the Singular Value…
Matrix factorization is a key tool in data analysis; its applications include recommender systems, correlation analysis, signal processing, among others. Binary matrices are a particular case which has received significant attention for…
Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated.It is proved that such an m\times n matrix almost surely has a constant…
Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy…
We investigated some difficulties that students often face when studying linear algebra at the undergraduate level, and identified some common mistakes and difficulties they often encountered when dealing with topics that require…